Editing Length of a module (section)

=== Finite dimensional vector spaces ===
Any finite dimensional vector space <math>V</math> over a field <math>k</math> has a finite length. Given a basis <math>v_1,\ldots,v_n</math> there is the chain<math display=block>0 \subset \text{Span}_k(v_1) \subset \text{Span}_k(v_1,v_2) \subset \cdots \subset \text{Span}_k(v_1,\ldots, v_n) = V</math>which is of length <math>n</math>. It is maximal because given any chain,<math display=block>V_0 \subset \cdots \subset V_m</math>the dimension of each inclusion will increase by at least <math>1</math>. Therefore, its length and dimension coincide.